The new formulation should be included in WAQUA, Delft3D and similar hydrodynamic models. It leads to a more realistic simulation of the wind effects. The semi-implicit implementation should be used, since it leads to more stability. Unfortunately, it has not been possible to adjust the wind formulations within WAQUA during this research.
Therefore, the real improvements of the IJsselmeer-model on the Ketelmeer-area as a result of this new formulation cannot be shown. However, the simplified 2D Ketelmeer-schematization proves that in a similar situation, the new formulation leads to a decrease of the water level in shallow areas during severe storm, which is exactly what is desired in the IJsselmeer-model.
When the new formulation is implemented in a 3D model, the precise effects of the real flow velocity at the surface versus the depth-averaged velocity can be measured. The absence of this influence in depth-averaged models might be compensated by introducing an extra factor in these models.
Research on the exact value of α is needed. A constant value can be taken, but a relation with the sea state or a non-linear relation with the wind speed might be more accurate. The factor α gives the possibility of adjusting the influence of the flowing water on the wind stress. Therefore it can also be used as a fitting parameter, which - in contradiction to the drag coefficient - only affects areas with a significant flow velocity.
More research on the drag coefficient is also necessary. The coefficient is currently
still fitted for each simulation separately. Some computer programs allow a space vary-ing drag coefficient. In very shallow areas often a lower coefficient is used. The new formulation may give better results on these shallow areas without adjusting the drag coefficient. Therefore, new fitting of the drag coefficient might be necessary. If the new formulation is used in existing models, users should be aware of this.
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Appendix A
Fortran code for the 1D-model
module arrays
implicit doubleprecision(a-h,o-z)
real*8, dimension(:), allocatable :: s1,s0,u1,u0,qp1,qp0,withs,withu,&
q1,q0,dp,h,mu,hu,wetss,wetsu,rhsu,wtest,a,b,c,d,au,bu,cu,ru,ua,du,se,ue,urel real*8, dimension(:,:), allocatable :: r,pz,apz,cpz,wa,w0,w1
integer, dimension(:), allocatable :: utype integer :: dtout1,dtout2
alpha=1.0 !correction factor for wind at 10 metres height to wind at surface qinput=1.0
sboundary=0.0
dmout=1 !for output: in x-direction space between writing of output point.
dtout1=10 !for output: number of timesteps to skip for writing history-output again.
dtout2=100 !for output: number of timesteps to skip for writing map-output again.
station1=(mmax-1)/2 station2=3*(mmax-1)/4 call bathymetry call initialise call output1
write(*,’(’’ Which wind stress formulation do you wish to use? ’’)’) write(*,’(’’ 1. Wind 1 - Common wind stress ’’)’)
write(*,’(’’ 2. Wind 2 - Relative wind velocity, explicit ’’)’)
write(*,’(’’ 3. Wind 3 - Relative wind velocity, implicit (alfa=1)’’)’)
write(*,’(’’ 4. Wind 3 - Relative wind velocity, implicit + alfa (correction for u10) ’’)’) 111 read *,wind
nenergy=0
write (*,’(’’Wrong wind-type, try again:’’)’) goto 111
cu(m)=0.0
PAUSE ’press key to continue’
contains
enddo
do m=1,mmax-1,dmout ! (write all depth-points in depth.map) x=m*dx
write(8,’(7e12.4)’) x, dp(m) enddo
write(8,’(7e12.4)’) x, dp(m)
write(9,’(7e12.4)’,advance=’no’) nt ! (1st cell contains #timesteps) do m=1,mmax-1,dmout ! (first row of alldata.his contains the x values)
x=m*dx
write(9,’(7e12.4)’,advance=’no’) s1(m) ! (write initial values) enddo
write(9,’(7e12.4)’,advance=’no’) t ! (first column contains curtime) do m=1,mmax-1,dmout
write(9,’(7e12.4)’,advance=’no’) s1(m) ! (other columns contain waterlevels) enddo
Appendix B
Source code 2D-model
C
C WIND AND BOTTOM FRICTION C
C End parameters for wind C
Appendix C
Derivation of the numerical parameters (1D)
Wind 1
is approximated by the following numerical solution:
uk+1m − ukm This will be rewritten in the form
uk+1m = αm+ βmζm+1k+1 + γmζmk+1
Wind 2
Now the momentum equation is
∂u
and is approximated by the following numerical solution:
uk+1m − ukm
Wind 3
Now the momentum equation is
∂u
which is approximated by the following numerical solution:
uk+1m − ukm
Wind 3 with alpha
Now the momentum equation is
∂u
which is approximated by the following numerical solution:
uk+1m − ukm